$ D = \left[\begin{array}{rrr}2 & -2 & 5 \\ 1 & -2 & 1\end{array}\right]$ $ F = \left[\begin{array}{rr}-2 & 5 \\ -1 & 5 \\ 5 & 2\end{array}\right]$ What is $ D F$ ?
Answer: Because $ D$ has dimensions $(2\times3)$ and $ F$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ D F = \left[\begin{array}{rrr}{2} & {-2} & {5} \\ {1} & {-2} & {1}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{5} \\ {-1} & \color{#DF0030}{5} \\ {5} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ D$ , with the corresponding elements in column $j$ of the second matrix, $ F$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ D$ with the first element in ${\text{column }1}$ of $ F$ , then multiply the second element in ${\text{row }1}$ of $ D$ with the second element in ${\text{column }1}$ of $ F$ , and so on. Add the products together. $ \left[\begin{array}{rr}{2}\cdot{-2}+{-2}\cdot{-1}+{5}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ D$ with the corresponding elements in ${\text{column }1}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-2}+{-2}\cdot{-1}+{5}\cdot{5} & ? \\ {1}\cdot{-2}+{-2}\cdot{-1}+{1}\cdot{5} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ D$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ F$ and add the products together. $ \left[\begin{array}{rr}{2}\cdot{-2}+{-2}\cdot{-1}+{5}\cdot{5} & {2}\cdot\color{#DF0030}{5}+{-2}\cdot\color{#DF0030}{5}+{5}\cdot\color{#DF0030}{2} \\ {1}\cdot{-2}+{-2}\cdot{-1}+{1}\cdot{5} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{2}\cdot{-2}+{-2}\cdot{-1}+{5}\cdot{5} & {2}\cdot\color{#DF0030}{5}+{-2}\cdot\color{#DF0030}{5}+{5}\cdot\color{#DF0030}{2} \\ {1}\cdot{-2}+{-2}\cdot{-1}+{1}\cdot{5} & {1}\cdot\color{#DF0030}{5}+{-2}\cdot\color{#DF0030}{5}+{1}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}23 & 10 \\ 5 & -3\end{array}\right] $